Point dividing crowd

In mathematics, a point-separating set is a set of functions in a given space, so that two points in the space can be distinguished on the basis of their function values ​​with regard to these functions. The term is used in general topology and functional analysis .

definition

Be a lot. A lot of functions with domain 's points separating if for any two elements with a function exists, so . ${\displaystyle X}$ ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle a,b\in X}$ ${\displaystyle a\neq b}$ ${\displaystyle f\in H}$ ${\displaystyle f(a)\neq f(b)}$

use

Again, be a set and a set of functions . You can now see the evaluation image ${\displaystyle X}$ ${\displaystyle H}$ ${\displaystyle X}$

${\displaystyle e\colon X\to \prod _{f\in H}\operatorname {codom} (f)}$

by defining ( be the target set of ). This is injective if and only if it is point-separating. ${\displaystyle e(x)_{f}=f(x)}$ ${\displaystyle \operatorname {codom} (f)}$ ${\displaystyle f}$ ${\displaystyle H}$

If a topological space and the set of all -valent continuous functions are on , then the conclusion of the picture is the Stone-Čech compactification of . Is point-separating (that is, it is a complete Hausdorff space ), so an identification of the set with a subset of the Stone-Čech compactification provides . ${\displaystyle X}$ ${\displaystyle H}$ ${\displaystyle [0,1]}$ ${\displaystyle X}$ ${\displaystyle e}$ ${\displaystyle X}$ ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle e}$ ${\displaystyle X}$

More generally, let be an arbitrary set of functions in topological spaces. The evaluation mapping is an embedding exactly when the initial topology is relevant and separating points. This initial topology is also called weak topology with regard to , especially in functional analysis, when there is a set of linear functionals on a vector space . If the target space of every function is in a Hausdorff space, then the weak topology is Hausdorffian with respect to and only if is point-separating. If a set of linear functionals is on a vector space, the point separation and thus the Hausdorff property of the weak topology can be characterized by the condition that ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle H}$

${\displaystyle \bigcap _{f\in H}\ker(f)=\{0\}}$

applies. In particular, it follows from Hahn-Banach's theorem that the set of all continuous linear functionals on a locally convex Hausdorff space is point-separating and thus the weak topology on such a space is Hausdorffian.

The Stone-Weierstrass theorem provides that a subalgebra of the algebra of functions on a locally compact Hausdorff space if and dense in is if it is divisive points and no point always on the maps. ${\displaystyle C_{0}}$ ${\displaystyle X}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle 0}$

Individual evidence

1. ↑ Point-dividing crowd . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. tape ? . Spectrum Akademischer Verlag, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 , p. ? .
2. ^ Nicolas Bourbaki : Topologie Générale (=  Éléments de mathématique ). Springer, Berlin 2007, ISBN 3-540-33936-1 , chap. 9 , p. 9 .
3. ↑ Stephen Willard: General Topology . Addison-Wesley , 1970, pp. 56 .
4. ↑ Bourbaki: Topology Générale. P. 10.
5. ^ Walter Rudin : Functional Analysis . McGraw-Hill, New York 1991, ISBN 0-07-054236-8 , pp. 60, 63 .